Optimal. Leaf size=29 \[ \frac {a x \log (x)}{\sqrt {c x^2}}+\frac {b x^2}{\sqrt {c x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {15, 43} \begin {gather*} \frac {a x \log (x)}{\sqrt {c x^2}}+\frac {b x^2}{\sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 43
Rubi steps
\begin {align*} \int \frac {a+b x}{\sqrt {c x^2}} \, dx &=\frac {x \int \frac {a+b x}{x} \, dx}{\sqrt {c x^2}}\\ &=\frac {x \int \left (b+\frac {a}{x}\right ) \, dx}{\sqrt {c x^2}}\\ &=\frac {b x^2}{\sqrt {c x^2}}+\frac {a x \log (x)}{\sqrt {c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 19, normalized size = 0.66 \begin {gather*} \frac {x (a \log (x)+b x)}{\sqrt {c x^2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 26, normalized size = 0.90 \begin {gather*} \sqrt {c x^2} \left (\frac {a \log (x)}{c x}+\frac {b}{c}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 22, normalized size = 0.76 \begin {gather*} \frac {\sqrt {c x^{2}} {\left (b x + a \log \relax (x)\right )}}{c x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.32, size = 35, normalized size = 1.21 \begin {gather*} -\frac {a \log \left ({\left | -\sqrt {c} x + \sqrt {c x^{2}} \right |}\right )}{\sqrt {c}} + \frac {\sqrt {c x^{2}} b}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 18, normalized size = 0.62 \begin {gather*} \frac {\left (a \ln \relax (x )+b x \right ) x}{\sqrt {c \,x^{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 20, normalized size = 0.69 \begin {gather*} \frac {a \log \relax (x)}{\sqrt {c}} + \frac {\sqrt {c x^{2}} b}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.51, size = 17, normalized size = 0.59 \begin {gather*} \frac {b\,\relax |x|+a\,\ln \left (c\,x\right )\,\mathrm {sign}\relax (x)}{\sqrt {c}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b x}{\sqrt {c x^{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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